Final answer:
To solve the homogeneous differential equation by substitution, express y in terms of ux, differentiate, and substitute into the original equation. Separate variables and integrate both sides to find u as a function of x, and then substitute back to express y in terms of x.
Step-by-step explanation:
To solve the given homogeneous differential equation dy/dx = y - xy + x using the substitution y = ux, first express y as ux and differentiate both sides with respect to x:
Since y = ux, we then have dy/dx = u + x(du/dx).
Substitute y and dy/dx in the original equation:
u + x(du/dx) = ux - ux^2 + x
Simplify and group like terms to obtain an equation involving u and x:
x(du/dx) = x - u^2x^2,
which simplifies further to:
du/dx = 1/x - u^2x.
This equation can be solved by separation of variables:
du/(1 - u^2x^2) = dx/x.
Integrating both sides gives:
½ ln |1 - u^2x^2| = ln|x| + C,
where C is the constant of integration. Solving for u and then substituting back for y yields the general solution of the differential equation.